The Direct Product Poisson Structure And Integrability For The High Order Restricted System Of AKNS Hierarchy

Based on the AKNS eigenvalue problem, with the help of the non-linearization approach of eigenvable problem, we obtain a finite-dimensional Hamilton system with direct product Poisson structure. And its integrability is proved by the generating function method. Further,the reduction on the common level sets of Casimir function is also introuduced by suitable coordinate transformation.